Bottom Line:
In a previous study, we defined exchangeability for blocks of data, as opposed to each datum individually, then allowing permutations to happen within block, or the blocks as a whole to be permuted.Here we extend that notion to allow blocks to be nested, in a hierarchical, multi-level definition.The strategy is compatible with heteroscedasticity and variance groups, and can be used with permutations, sign flippings, or both combined.

f0015: The multi-level definition of blocks allows more complex relationships between observations. Left: Three blocks of identical structure (2nd column) can be shuffled as a whole (as indicated by the positive indices in the 1st column); within each (3rd column), only two out of their three constituting observations can be swapped (1 and 2, 4 and 5, and 7 and 8), whereas the third on each (3, 6 and 9) cannot; levels for these last branches are completed with blocks for which the sign has no meaning (in black), as they remain unaltered towards the next level (4th column), and represent no actual branching. In the visual representation, these black blocks are shown as small black dots on continuous branches. This example could represent 3 sets of siblings, each composed of a pair of monozygotic twins and a third non-twin. Centre: An example showing that it is possible to mix types of blocks in the same level (2nd column). As shown, the first two blocks in the 2nd column cannot be swapped despite similar coding, and neither of these can be permuted with the third, which has a different structure consisting of three observations (7, 8 and 9) that can be shuffled freely. This example could represent 3 sets of siblings, the first a pair of monozygotic twins and a non-twin, the second a pair of dizygotic twins and a non-twin (if certain environmental effects are considered), and the third a set of three non-twin siblings. Right: The same notation can also accommodate simple designs. Here all 9 observations can be permuted without restrictions on exchangeability.

Mentions:
Using the tree diagram, it becomes clear that the terms “within-block” and “whole-block”, that have been used so far to describe exchangeability and permutation strategies, become no longer necessary, as either the branches can be shuffled, or they cannot. It is also helpful in emphasising that more complicated designs can be considered using multi-level blocks, in which even the distinction between within- and whole-block is softened, as each level in the multi-column notation is not restricted to contain purely positive or negative indices restricting (or not) the shuffling of their constituent sub-blocks (branches). These can be present alongside each other if immediately below a level in which shuffling is not allowed, such that some branches may be allowed to be shuffled, whereas others are not. It may also be the case that some levels need to be included in the notation only so that the number of levels remains the same across all branches of the tree, from the top node to the most distal (leaves), without affecting the construction of , but ensuring that the notation can be stored, without gaps, in a two-dimensional array; in the visual representation these are shown as small, sign-less, black nodes. Fig. 3 (left and centre) exemplifies these cases. Although the multi-column notation and the corresponding tree can become very complex, the simple, unrestricted exchangeability can also be accommodated, as shown in Fig. 3 (right).

f0015: The multi-level definition of blocks allows more complex relationships between observations. Left: Three blocks of identical structure (2nd column) can be shuffled as a whole (as indicated by the positive indices in the 1st column); within each (3rd column), only two out of their three constituting observations can be swapped (1 and 2, 4 and 5, and 7 and 8), whereas the third on each (3, 6 and 9) cannot; levels for these last branches are completed with blocks for which the sign has no meaning (in black), as they remain unaltered towards the next level (4th column), and represent no actual branching. In the visual representation, these black blocks are shown as small black dots on continuous branches. This example could represent 3 sets of siblings, each composed of a pair of monozygotic twins and a third non-twin. Centre: An example showing that it is possible to mix types of blocks in the same level (2nd column). As shown, the first two blocks in the 2nd column cannot be swapped despite similar coding, and neither of these can be permuted with the third, which has a different structure consisting of three observations (7, 8 and 9) that can be shuffled freely. This example could represent 3 sets of siblings, the first a pair of monozygotic twins and a non-twin, the second a pair of dizygotic twins and a non-twin (if certain environmental effects are considered), and the third a set of three non-twin siblings. Right: The same notation can also accommodate simple designs. Here all 9 observations can be permuted without restrictions on exchangeability.

Mentions:
Using the tree diagram, it becomes clear that the terms “within-block” and “whole-block”, that have been used so far to describe exchangeability and permutation strategies, become no longer necessary, as either the branches can be shuffled, or they cannot. It is also helpful in emphasising that more complicated designs can be considered using multi-level blocks, in which even the distinction between within- and whole-block is softened, as each level in the multi-column notation is not restricted to contain purely positive or negative indices restricting (or not) the shuffling of their constituent sub-blocks (branches). These can be present alongside each other if immediately below a level in which shuffling is not allowed, such that some branches may be allowed to be shuffled, whereas others are not. It may also be the case that some levels need to be included in the notation only so that the number of levels remains the same across all branches of the tree, from the top node to the most distal (leaves), without affecting the construction of , but ensuring that the notation can be stored, without gaps, in a two-dimensional array; in the visual representation these are shown as small, sign-less, black nodes. Fig. 3 (left and centre) exemplifies these cases. Although the multi-column notation and the corresponding tree can become very complex, the simple, unrestricted exchangeability can also be accommodated, as shown in Fig. 3 (right).

Bottom Line:
In a previous study, we defined exchangeability for blocks of data, as opposed to each datum individually, then allowing permutations to happen within block, or the blocks as a whole to be permuted.Here we extend that notion to allow blocks to be nested, in a hierarchical, multi-level definition.The strategy is compatible with heteroscedasticity and variance groups, and can be used with permutations, sign flippings, or both combined.